Good Reduction of K3 Surfaces

TL;DR

This paper establishes conditions under which K3 surfaces over local fields have good reduction after unramified extensions, linking Galois actions on cohomology to geometric properties, with applications to tame extensions and Abelian varieties.

Contribution

It proves that unramified Galois actions on second ℓ-adic cohomology imply good reduction after unramified extensions for K3 surfaces, and addresses semi-stable flops and group actions in mixed characteristic.

Findings

Unramified Galois action implies good reduction after unramified extension.

Examples show the necessity of unramified extensions.

Applications to tame extensions and Kuga-Satake Abelian varieties.

Abstract

Let $K$ be the field of fractions of a local Henselian DVR with perfect residue field. Assuming potential semi-stable reduction, we show that an unramified Galois-action on second $\ell$-adic cohomology of a K3 surface over $K$ implies that the surface has good reduction after a finite and unramified extension. We give examples where this unramified extension is really needed. Moreover, we give applications to good reduction after tame extensions and Kuga-Satake Abelian varieties. On our way, we settle existence and termination of certain semi-stable flops in mixed characteristic, and study group actions and their quotients on models of varieties.